In this Appendix, we briefly list all the perturbed quantities in tensor, vector, and scalar perturbations and outline the perturbation equations for our specific dark energy model from the Generalized Proca theory, as in Refs. [1]}, [2]}, [3]}. We also adopt the method described in Ref. [1]}. Specifically, we first expand  (REF ) up to the second order in perturbations,
and then vary the second-order action
with respect to the perturbed quantities to arrive at the perturbation equations.
To perturb the gravity part of the action, we take
the perturbing line element [5]}, [6]},
\(ds^2=-(1+2\alpha )dt^2+2(\partial _i \chi + V_i)dtdx^i+a^2(t)(\delta _{ij}+h_{ij})dx^i dx^j,\) 
